An optimal transportation approach for assessing almost stochastic order

Abstract

When stochastic dominance $F{\le }_{st}G$ does not hold, we can improve agreement to stochastic order by suitably trimming both distributions. In this work we consider the $L_2-$Wasserstein distance, ${\mathcal{W}}_2$, to stochastic order of these trimmed versions. Our characterization for that distance naturally leads to consider a ${\mathcal{W}}_2$-based index of disagreement with stochastic order, ${\epsilon }_{{\mathcal{W}}_2}(F,G)$. We provide asymptotic results allowing to test $H_0:{\epsilon }_{{\mathcal{W}}_2}(F,G)\ge {\epsilon }_0$ vs $H_a:{\epsilon }_{{\mathcal{W}}_2}(F,G)<{\epsilon }_0$, that, under rejection, would give statistical guarantee of almost stochastic dominance. We include a simulation study showing a good performance of the index under the normal model.